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April 2004 A note on the exponential diophantine equation $a^x + b^y = c^z$
Maohua Le
Proc. Japan Acad. Ser. A Math. Sci. 80(4): 21-23 (April 2004). DOI: 10.3792/pjaa.80.21

Abstract

Let $a$, $b$, $c$ be fixed coprime positive integers. In this paper we prove that if $b \equiv 3 \pmod{4}$, $a \equiv -1 \pmod{b^{2l}}$, $a^2 + b^{2l-1} = c$ and $c$ is odd, where $l$ is a positive integer, then the equation $a^x + b^y = c^z$ has only the positive integer solution $(x,y,z) = (2,2l-1,1)$.

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Maohua Le. "A note on the exponential diophantine equation $a^x + b^y = c^z$." Proc. Japan Acad. Ser. A Math. Sci. 80 (4) 21 - 23, April 2004. https://doi.org/10.3792/pjaa.80.21

Information

Published: April 2004
First available in Project Euclid: 18 May 2005

zbMATH: 1050.11040
MathSciNet: MR2055070
Digital Object Identifier: 10.3792/pjaa.80.21

Subjects:
Primary: 11D61

Keywords: exponential diophantine equations , primitive divisors of Lucas numbers

Rights: Copyright © 2004 The Japan Academy

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Vol.80 • No. 4 • April 2004
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